Mix zucchini and salt in a strainer; set a weighted plate on top; let drain for 30 minutes. Transfer zucchini to a cheese cloth; squeeze out liquid. Mix zucchini, parsley, cheese, bread crumbs, onions, and egg in a bowl. Season with pepper and cayenne; divide mixture into 12 balls. Press balls into ¾"-thick patties.

Pour oil into a 4-qt. pot to a depth of 2"; heat over medium-high heat until a deep-fry thermometer reads 315°. Working in 2 batches, fry patties until browned and crisp, 5–6 minutes. Using a slotted spoon, transfer fritters to paper towels.